Properties

Label 259182.fh
Number of curves $6$
Conductor $259182$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("fh1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 259182.fh

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
259182.fh1 259182fh6 \([1, -1, 1, -14939559779, -702833547127269]\) \(285531136548675601769470657/17941034271597192\) \(23170271092499019938343048\) \([2]\) \(314572800\) \(4.3274\)  
259182.fh2 259182fh4 \([1, -1, 1, -935498939, -10937714017557]\) \(70108386184777836280897/552468975892674624\) \(713495986231622458635118656\) \([2, 2]\) \(157286400\) \(3.9808\)  
259182.fh3 259182fh5 \([1, -1, 1, -318645779, -25146803187525]\) \(-2770540998624539614657/209924951154647363208\) \(-271111350310116635073441084552\) \([2]\) \(314572800\) \(4.3274\)  
259182.fh4 259182fh2 \([1, -1, 1, -98798459, 95018511723]\) \(82582985847542515777/44772582831427584\) \(57822356616188051279056896\) \([2, 2]\) \(78643200\) \(3.6342\)  
259182.fh5 259182fh1 \([1, -1, 1, -76495739, 257212812651]\) \(38331145780597164097/55468445663232\) \(71635720864281089015808\) \([2]\) \(39321600\) \(3.2877\) \(\Gamma_0(N)\)-optimal
259182.fh6 259182fh3 \([1, -1, 1, 381058501, 747048148971]\) \(4738217997934888496063/2928751705237796928\) \(-3782389016468744260638599232\) \([2]\) \(157286400\) \(3.9808\)  

Rank

sage: E.rank()
 

The elliptic curves in class 259182.fh have rank \(1\).

Complex multiplication

The elliptic curves in class 259182.fh do not have complex multiplication.

Modular form 259182.2.a.fh

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{4} + 2q^{5} - q^{7} + q^{8} + 2q^{10} + 2q^{13} - q^{14} + q^{16} + q^{17} - 4q^{19} + O(q^{20})\)  Toggle raw display

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.